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Consider the initial valuc problem t2y' + 3ty 6t3 _ 0, 9(0) _ 1Using the improved Euler method, approximate y(2) using a step size ofh = %....

Question

Consider the initial valuc problem t2y' + 3ty 6t3 _ 0, 9(0) _ 1Using the improved Euler method, approximate y(2) using a step size ofh = %.

Consider the initial valuc problem t2y' + 3ty 6t3 _ 0, 9(0) _ 1 Using the improved Euler method, approximate y(2) using a step size ofh = %.



Answers

Use Euler's method with step size 0.1 to estimate $y(0.5),$ where $y(x)$ is the solution of the initial-value problem $y^{\prime}=y+x y, y(0)=1$

Given an initial value problem, we were asked to use Oilers method with a specific steps size to find the value of a solution. At some point X were given differential equation y prime equals y plus X y with initial value. Y zero equals one and step size is 0.1, and you want to find why of 0.5 approximately. So since step sizes 0.1, we're starting at X zero equals zero. Well, expect five steps. We have x 00 Why zero is going to be y over x zero, which is one x one is going to be X. You're a pulsar step size just 10.1 and why one is going to be y zero plus our step size times our function y plus x y evaluated at x zero y zero just simply one. This is equal to 1.1 x two is going to be x one plus or step size 0.2. Why to is why one plus our step size times are function y plus X y Evaluated at x one y one 1.1 plus 0.1 times 1.1 This is equal to 1.2 to 1 I x three is equal to x two plus or step size, which is 20.3. Why three z people, too? Why two You're gonna start just writing out why two instead of the full value plus our step size 0.1 times that I have are function at x two y two, which is why two plus 0.2 times why, too. This is equal to 1.36752 x for his X three plus or step size. Just point for and wife for is why three plus or step size times a function evaluated at x three y three Just why three plus 30.3 times y three which is equal to 1.54 five to nine 76 approximately. And finally we have our step five x five is equal 2.4 plus 14.5 We have that. Why of 0.5 Under this approximation is about equal to why five, which is equal to y four plus our step size times the function evaluated at X for y for so wife work plus points for why, for which is equal to approximately Yeah 1.76 16 393 And if you use the store function on a calculator or have a calculator, which saves more digits, you can get a more accurate answer than this. I think this is pretty accurate.

Were given initial value problem. We were asked to use Oilers method to approximate the value of the solution. At a given value of X, we're given the differential equation y prime goes X squared y minus 1/2 y squared with initial value. Why it zero equals one. And the steps eyes were asked to use his each equals point to you were asked to find the value of why of one. Our initial value tells us that X zero is equal to zero and y zero. His wife of X zero, which is one x one is x zero plus er step size, which is 00.2 and why one is equal to why zero, which is one plus our step size point to times the value of our function. X squared y minus 1/2 y squared at 01 just simply negative. 1/2 1 squared is equal to one plus point to times negative. 0.5 is equal to point 0.9. X two is going to be x one, plus her step function by step just point for And why to is why one 0.9 plus step size 22 times Value of our function at 0.2 point nine. This is 0.2 squared times 0.9 minus 1/2 times 0.9 squared you point 82 six to x three is going to be X two plus or step size. So 20.6 in why three it's going to be. Instead of writing out why? To simply write Why, too, Plus our steps. Signs point to terms of a ver functions at point for why to 4.4 squared y two minus 1/2. Why two squared? This is too 0.78 for three 778 Approximately X four is equal to 0.6 plus or step size or 0.8. And why four? It's going to be equal to buy three plus our steps eyes point to times are function evaluated x three y three, which is 0.6 Swear times Wife three minus 1/2. Why three squared, which is equal to 0.779 32 81 Approximately. Finally, we have that X five is going to be X for pleasure. Step size, which is one in Y five, is equal to y for plus step size 0.2 times or function evaluated at X for y for just point it squared. Why four minus one? Have why four squared? Which is equal to approximately point 81 83 469 And we have that y five. This is approximately equal to why have one. So our answer is going 8183 for 69 or 0.81 83 for seven after rounding.

Okay, so it is given that each is equal to zero. Pulling to accept zero Z with zero y zero is equal to one on dhe F off X comma y is equal to X y minus X squared. So we could do why of 0.2 boy, 0.2 is equal to one plus 0.2 multiplied by zero times one minus zero to the seat to one. Thanks. You have a Y 0.4, which is equal to one plus 0.2 multiplied by 0.2 Minuses. Your 0.4 This is equal to 1.32 Next we have y 0.6, which is equal to 11 of Phyllis out again. But you should be able to get the pattern by that. This is 0.6 11 1.3 to 0.2 times two plus the airplane four times 1.332 times your foot foreswear. So I think there is 1.0 age, too. Why of 0.8 is equal to 1.14 and why 1.0 is equal to 1.1 night before

Okay, so we're given a differential equation. Were given an initial value or given a step size h of 0.1. And we have to use Oilers method. Find the next five values of the solution. And I've written in blue here are recursive relationship that we use for oilers formulas. Um, specific for this problem. Ah, we can call this y en plus H and R function is given, and we'll write it with ends accident over. Why in instead of just x over? Why, okay, and then from here we just have a lot of plugging jug to do. So let's start with an equal zero when in a zero, we'd have x one equals x zero lost age and are zero is given. Zero is zero and are y zero is negative one. Those are our initial conditions. So we have zero, which is zero plus h 2.1 and for why one we would have y zero plus h x zero over y zero. And again, all this stuff is given why zero is negative one plus h is there 0.1 x zero ver y zero would be zero over minus one and you can see here that this will be negative One. Okay, so x one is 10.1. Why one is negative one. All right. We'll continue on for any Kools. One equals one X two is x one plus each. Carry some information. Forward or x one and R Y one from our previous calculations, X one, this 10.1. Why won't his negative one Okay, so our X 10.1 plus R h, which is 0.1 is gonna 0.2 are y two, you know, That's why one plus h this time be ex foreign over. Why one we can fill in here. Why one is negative. One R H 0.1. Excellent. 0.1. Why want his negative one? Okay. And with some math, we will get This is negative. 1.1 All right, let's carry that forward. So here we just calculated x to excuse me. RX two was 0.2. And are y two was negative. 1.1 So let's do one more in detail. So happy at n equals two. That tells us that x three is X two plus h and I think I made a mistake here? Um, X two is 0.2. Okay, X rays, X two plus h X two is your 20.2 plus our age of 0.1 at this 0.3. And why three is our Y two plus each X two over y two. All the stuff we calculated on the previous step y two is minus 1.1. We know where h you know where. X two. You know why? Too little bit of math minus 1.0 to 98 case. Whether it's our extra or y three, we're gonna need X for y for we're gonna need x five y five and I won't write out all the details follows the same way that we iterated through the 1st 3 Ah, for these would have zero point for minus 1.589 x 50.5. Image of minus 1.14 to 3. Okay. And those would be your final answers


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